Abstract

We define and study new classifications of qcb\(_0\)-spaces based on the idea to measure the complexity of their bases. The new classifications complement those given by the hierarchies of qcb\(_0\)-spaces introduced in [7, 8] and provide new tools to investigate non-countably based qcb\(_0\)-spaces. As a by-product, we show that there is no universal qcb\(_0\)-space and establish several apparently new properties of the Kleene-Kreisel continuous functionals of countable types.

M. Schröder—Supported by FWF research project “Definability and computability” and by DFG project Zi 1009/4-1.

V. Selivanov—Supported by the DFG Mercator professorship at the University of Würzburg, by the RFBR-FWF project “Definability and computability”, by RFBR project 13-01-00015a, and by 7th EU IRSES project 294962 (COMPUTAL).